Miskolc Mathematical Notes HU e-ISSN 1787-2413 Vol. 19 (2018), No. 1, pp. 63–76 DOI: 10.18514/MMN.2018.1484 PURELY BAER MODULES AND PURELY RICKART MODULES S. EBRAHIMI ATANI, M. KHORAMDEL, AND S. DOLATI PISHHESARI Received 30 December, 2014 Abstract. We introduce the notions of purely Baer and purely Rickart modules. We provide several characterizations and investigate properties of each of these concepts. We provide new characterizations of several well-known classes of rings in terms of purely Baer and purely Rick- art modules. It is shown that R is a von Neumann regular ring iff every right R-module is purely Baer (purely Rickart). Also, we prove R is left semihereditary iff every (finitely generated) free right R-module is purely Baer. Examples illustrating the results are presented. 2010 Mathematics Subject Classification: 16E50; 16E60; 16D80 Keywords: Baer modules, purely Baer modules, Rickart modules, purely Rickart modules 1. INTRODUCTION The notions of Rickart and Baer rings have their roots in functional analysis, with close links to C -algebras and von Neumann algebras. Kaplansky introduced Baer rings to abstract various properties of AW -algebras and von Neumann algebras and complete -regular rings in [11]. Motivated by Kaplansky’s work on Baer rings, the notion of Rickart rings appeared in Maeda [15] and was further studied by Hattori [9] and other authors. A ring R is called Baer (resp. right Rickart (or p.p.)) if the right annihilator of any nonempty subset (resp. any single element) of R is generated by an idempotent, as a right ideal of R. It is well known that Baer rings and Rickart rings play an important role in providing a rich supply of idempotents and hence in the structure theory for rings. Recently, the notions of Baer and Rickart rings and their generalizations were extended and studied in a general module-theoretic setting [1, 13, 14, 16–18]. A module is called extending if every its submodule is essential in a direct sum- mand. In [6], the notion of extending generalized to purely extending by replacing “direct summand” with “pure submodule”. In [4], basic characterizations of purely extending modules are given. Motivated by the notions of Baer and purely extending modules, we introduce the notion of purely Baer modules. In Section 3, we investigate purely Baer modules and give some results related to them. It is shown that a direct summand of a purely Baer module is purely Baer. Our focus, in this section, is on the question: When are (free) c 2018 Miskolc University Press 64 S. EBRAHIMI ATANI, M. KHORAMDEL, AND S. DOLATI PISHHESARI right R-modules over a ring R purely Baer? We obtain characterizations of well- known classes of rings, in terms of free purely Baer modules over them. We show that the class of rings for which every (resp. free) module is purely Baer, is precisely that of (resp. left semihereditary) von Neumann regular rings. As an application of this, we prove that a commutative domain R is Prufer¨ if and only if every free R- module is purely Baer. Some characterizations of right purely Baer rings are given. It is known that the definition of Baer rings is left-right symmetric. However we show that the definition of purely Baer rings is asymmetric. In Section 4, the notion of purely Rickart modules is introduced. A characteriza- tion of von Neumann regular rings in terms of purely Rickart modules is given. It is shown that every free right R-module is purely Rickart if and only if every right ideal of R is flat. We provide some characterizations of right purely Rickart rings. Also we show that the definition of purely Rickart rings is left-right symmetric, however the definition of Rickart rings is asymmetric. 2. PRELIMINARIES Throughout this paper, R is an associative ring with identity and all modules are unitary. By MR (resp. RM ), we denote a right (resp. left) R-module and S I .I / D End.MR/ denotes the endomorphism ring of MR. M (resp. M ) stands for the direct product (resp. direct sum) of copies of M indexed by a set I . Let M be a right R-module, I a subset of R and X a subset of End.MR/. We write rM .X/ m D f 2 M xm 0 for all x X and rR.I / (resp: lR.I /) = the right (resp. left) annihilator I inW R. LetD M be a module2 g over a ring R. For submodules N and K of M , N K ess Ä denotes N is a submodule of K. I what follows, by ˚, and E.M / we denote, respectively, a module direct summand, an essentialÄ submoduleÄ and the injective hull of M . For a ring R, Matn.R/ denotes the ring of n n matrices over R. In the following, we recall some known notions and facts needed in the sequel. ' Definition 1. (1) A (short) exact sequence 0 K N M 0 of right R- W ! ! ! ! modules is said to be pure (exact) if R T is an exact sequence (of abelian groups) for any left R -module T . In this case,˝ we say that '.K/ is a pure submodule of N (see [12] and [19]). It is clear that every direct summand is a pure submodule. (2) A module M over a ring R will be called finitely presented if there exists an exact sequence 0 K F M 0 of R-modules, where F is free and both F and K are finitely! generated! (see! [12!] and [19]). (3) A right module M over a ring R is called divisible provided M x M for all regular elements x R (see [8]). D (4) A right R- module2 M is called pure injective, if M is injective with respect to every pure exact sequence of right R-modules. A ring R is called right pure semisimple if every right R-module is pure injective (see [22]). PURELY BAER MODULES AND PURELY RICKART MODULES 65 (5) A module MR is called FP-injective (or absolutely pure) if, for any finitely generated submodule K of a free right R-module F , every homomorphism K M extends to a map F M (see [22]). ! ! (6) A module MR is called regular, provided that each of its submodule is pure (see [5]). (7) A right R-module M is said to be Baer (resp. Rickart), if for any left ideal I of End.M / (resp. End.M /), rM .I / (resp. rM ./) is a direct summand of M (see [17], [18], [13] and2 [14]). (8) A module M is called purely extending if every submodule of M is essential in a pure submodule of M . Equivalently, every closed submodule is a pure submodule (see [4]). (9) A module is called torsionless if it can be embedded in a direct product of copies of the base ring (see [12]). We refer to [12] and [22] for the undefined notions in this article. Proposition 1 ([12, Corollary 4.86]). Let 0 K F M 0 be an exact sequence of R-modules with F flat. Then M isW flat! if and! only! if is! pure. Proposition 2 ([21, Lemma 2.2]). Let R be a ring and 0 K P M 0 an exact sequence of right R-modules with P projective. Then! the following! ! statements! are equivalent: (1) M is flat; (2) Given any x K, there exists a homomorphism g P K such that g.x/ x; 2 W ! D (3) Given any x1;:::;xn in K, there exists a homomorphism g P K such that W ! g.xi / xi for i 1;2;:::;n. D D Proposition 3 ([2, Theorem 4.1] ). For any ring R, the following statements are equivalent: (1) R is left semi-hereditary; (2) Every right ideal of R is flat, and the direct product of an arbitrary family of copies of R is flat as a right R-module; (2) Every torsion-less right R-module is flat. 3. PURELY BAER MODULES In this section, we investigate connections between purely Baer modules and vari- ous existing concepts and obtain some of their useful properties. Examples of purely Baer modules include semisimple modules, regular modules, Baer modules, nonsin- gular purely extending modules and modules over von Neumann regular rings. Definition 2. Let M be a right R-module and S EndR.M /. Then M is called D a purely Baer module, if rM .I / is a pure submodule of M for each left ideal I of S. A ring R is called (resp. left) right purely Baer, if (resp. lR.I /) rR.I / is a pure (resp. left) right ideal of R for each (resp. right) left ideal I of R. 66 S. EBRAHIMI ATANI, M. KHORAMDEL, AND S. DOLATI PISHHESARI First, we characterize right purely Baer rings in terms of cyclic torsionless mod- ules. Theorem 1. Let R be a ring. Then R is right purely Baer if and only if every cyclic torsionless right R-module is flat. Proof. It is known that a cyclic right R-module R=I is torsionless if and only if I rR.X/ for some subset X of R. Assume that R is right purely Baer. Let M beD a cyclic torsionless right R-module. As M R=I for some right ideal I of R, Š I rR.X/ for some X R. Since R is right purely Baer, I rR.J / is pure in RR, whereD J is a left ideal generated by X. Thus R=I is flat by PropositionD 1 and so M is flat.